Problem 15 - Advanced Engineering Mathematics

Question

Find the first partial derivatives of the given function. $$ z=5 x^{4} y^{3}-x^{2} y^{6}+6 x^{5}-4 y $$

Step-by-Step Solution

Answer

The partial derivatives are \( \frac{\partial z}{\partial x} = 20x^3 y^3 - 2x y^6 + 30x^4 \) and \( \frac{\partial z}{\partial y} = 15x^4 y^2 - 6x^2 y^5 - 4 \).

Step 1: Understand the given function

We are given the function \( z = 5x^4 y^3 - x^2 y^6 + 6x^5 - 4y \). We need to find the first partial derivatives with respect to \( x \) and \( y \).

Step 2: Differentiate with respect to x

To find the partial derivative of \( z \) with respect to \( x \), treat \( y \) as a constant while differentiating. 1. Differentiate \( 5x^4 y^3 \) with respect to \( x \): \( 20x^3 y^3 \).2. Differentiate \( -x^2 y^6 \) with respect to \( x \): \( -2x y^6 \).3. Differentiate \( 6x^5 \) with respect to \( x \): \( 30x^4 \).4. The term \( -4y \) has no \( x \), so its derivative is \( 0 \).Therefore, the partial derivative with respect to \( x \) is: \[ \frac{\partial z}{\partial x} = 20x^3 y^3 - 2x y^6 + 30x^4 \].

Step 3: Differentiate with respect to y

To find the partial derivative of \( z \) with respect to \( y \), treat \( x \) as a constant while differentiating. 1. Differentiate \( 5x^4 y^3 \) with respect to \( y \): \( 15x^4 y^2 \).2. Differentiate \( -x^2 y^6 \) with respect to \( y \): \( -6x^2 y^5 \).3. The term \( 6x^5 \) has no \( y \), so its derivative is \( 0 \).4. Differentiate \( -4y \) with respect to \( y \): \( -4 \).Therefore, the partial derivative with respect to \( y \) is: \[ \frac{\partial z}{\partial y} = 15x^4 y^2 - 6x^2 y^5 - 4 \].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves calculating the rate at which a quantity changes. This process is known as finding the derivative. In simpler terms, if you have a function, the derivative provides information on how the function's output varies as you change the input. For single-variable functions, differentiation is straightforward. You only have one variable to consider.

When you differentiate a function like \( y = x^2 \), you find \( \frac{dy}{dx} = 2x \). This tells you the slope or rate of change of the function at any point \( x \). Differentiation has broad applications, including in physics for calculating velocity and acceleration, and in economics for analyzing profit maximization.

Differentiation is truly a versatile tool that aids in describing how systems evolve or vary over time or space.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions involving more than one variable. It opens up a wealth of possibilities for modeling and analyzing real-world phenomena. The functions you encounter are like landscapes with hills (maxima), valleys (minima), and saddle points, rather than simple curves on a graph.

In multivariable calculus, we often deal with functions like \( z = f(x, y) \), which map two variables to a single output. This becomes useful when analyzing systems with more than one influencing factor, such as temperature over a region or economic output based on labor and capital.

• Gradient: A vector that points in the direction of the steepest ascent and plays a similar role to the derivative in single-variable calculus.
• Level curves: Contours representing points of equal function value, useful for visualizing multivariable functions.

Multivariable calculus allows us to find local optima and model complex systems, making it essential for disciplines such as engineering, physics, and economics.

Partial Derivative Rules

Partial derivatives extend the concept of single-variable differentiation to multivariable calculus. They measure how a function changes as one of the variables is altered, keeping the others constant. Imagine you are at a point in a landscape and move east or north, partial derivatives tell you the slope in each direction.

To find the partial derivative with respect to \( x \), denoted \( \frac{\partial z}{\partial x} \), keep \( y \) constant and differentiate the function as though it only depends on \( x \). Similarly, to find \( \frac{\partial z}{\partial y} \), consider \( x \) as a constant.

• Power rule: Differentiate functions like \( ax^n \) by multiplying the power \( n \) by the coefficient \( a \) and decrementing the power: \( n \cdot ax^{n-1} \).
• Constant multiple rule: Multiply the derivative by the constant factor if the function is multiplied by a constant.

These rules allow us to manage the complexity of differentiating multivariable functions. Partial derivatives are foundational in fields like physics, optimization, and machine learning, offering insights into how systems respond to changes in individual parameters.